Relative $K$-theory via 0-cycles in finite characteristic

TL;DR

This paper establishes a pro-isomorphism between additive higher Chow groups of relative 0-cycles with modulus and the relative $K$-theory of truncated polynomial rings over certain regular semi-local rings in characteristic $p \\ge 3$, resolving a key conjecture.

Contribution

The authors prove that the cycle class map induces a pro-isomorphism linking additive higher Chow groups and relative $K$-theory in the setting of regular semi-local rings over fields of characteristic $p \\ge 3$, confirming a conjecture.

Findings

Pro-isomorphism between additive higher Chow groups and relative $K$-theory.

Resolution of the problem relating 0-cycles with modulus to relative $K$-theory.

Validation of the cycle class map as an isomorphism in this context.

Abstract

Let $R$ be a regular semi-local ring, essentially of finite type over an infinite perfect field of characteristic $p \ge 3$. We show that the cycle class map with modulus from an earlier work of the authors induces a pro-isomorphism between the additive higher Chow groups of relative 0-cycles and the relative $K$-theory of truncated polynomial rings over $R$. This settles the problem of equating 0-cycles with modulus and relative $K$-theory of such rings via the cycle class map.